Integration and decompositions of weak$^*$-integrable multifunctions

Kazimierz Musiał

Displaying similar documents to “Integration and decompositions of weak $^{*}$ -integrable multifunctions”

Convergence of ap-Henstock-Kurzweil integral on locally compact spaces

Hemanta Kalita, Ravi P. Agarwal, Bipan Hazarika (2025)

Czechoslovak Mathematical Journal

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We introduce an ap-Henstock-Kurzweil type integral with a non-atomic Radon measure and prove the Saks-Henstock type lemma. The monotone convergence theorem, $μ_{ap}$ -Henstock-Kurzweil equi-integrability, and uniformly strong Lusin condition are discussed.

Banach-valued Henstock-Kurzweil integrable functions are McShane integrable on a portion

Tuo-Yeong Lee (2005)

Mathematica Bohemica

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It is shown that a Banach-valued Henstock-Kurzweil integrable function on an $m$ -dimensional compact interval is McShane integrable on a portion of the interval. As a consequence, there exist a non-Perron integrable function $f {[0, 1]}^{2} ⟶ ℝ$ and a continuous function $F {[0, 1]}^{2} ⟶ ℝ$ such that $(¶) \int_{0}^{x} \{(¶) \int_{0}^{y} f (u, v) d v\} d u = (¶) \int_{0}^{y} \{(¶) \int_{0}^{x} f (u, v) d u\} d v = F (x, y)$ for all $(x, y) \in {[0, 1]}^{2}$ .

On a generalization of Henstock-Kurzweil integrals

Jan Malý, Kristýna Kuncová (2019)

Mathematica Bohemica

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We study a scale of integrals on the real line motivated by the $M C_{α}$ integral by Ball and Preiss and some recent multidimensional constructions of integral. These integrals are non-absolutely convergent and contain the Henstock-Kurzweil integral. Most of the results are of comparison nature. Further, we show that our indefinite integrals are a.e. approximately differentiable. An example of approximate discontinuity of an indefinite integral is also presented.

The Denjoy extension of the Riemann and McShane integrals

Jae Myung Park (2000)

Czechoslovak Mathematical Journal

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In this paper we study the Denjoy-Riemann and Denjoy-McShane integrals of functions mapping an interval $[a, b]$ into a Banach space $X .$ It is shown that a Denjoy-Bochner integrable function on $[a, b]$ is Denjoy-Riemann integrable on $[a, b]$ , that a Denjoy-Riemann integrable function on $[a, b]$ is Denjoy-McShane integrable on $[a, b]$ and that a Denjoy-McShane integrable function on $[a, b]$ is Denjoy-Pettis integrable on $[a, b] .$ In addition, it is shown that for spaces that do not contain a copy of $c_{0}$ , a measurable Denjoy-McShane...

The $L^{r}$ Henstock-Kurzweil integral

Paul M. Musial, Yoram Sagher (2004)

Studia Mathematica

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We present a method of integration along the lines of the Henstock-Kurzweil integral. All $L^{r}$ -derivatives are integrable in this method.

Characterizations of Kurzweil-Henstock-Pettis integrable functions

L. Di Piazza, K. Musiał (2006)

Studia Mathematica

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We prove that several results of Talagrand proved for the Pettis integral also hold for the Kurzweil-Henstock-Pettis integral. In particular the Kurzweil-Henstock-Pettis integrability can be characterized by cores of the functions and by properties of suitable operators defined by integrands.

Integro-differential equations on time scales with Henstock-Kurzweil delta integrals

Aneta Sikorska-Nowak (2011)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper we prove existence theorems for integro - differential equations $x^{Δ} (t) = f (t, x (t), \int ₀^{t} k (t, s, x (s)) Δ s)$ , t ∈ Iₐ = [0,a] ∩ T, a ∈ R₊, x(0) = x₀ where T denotes a time scale (nonempty closed subset of real numbers R), Iₐ is a time scale interval. Functions f,k are Carathéodory functions with values in a Banach space E and the integral is taken in the sense of Henstock-Kurzweil delta integral, which generalizes the Henstock-Kurzweil integral. Additionally, functions f and k satisfy some boundary conditions and...

A full characterization of multipliers for the strong $ρ$ -integral in the euclidean space

Lee Tuo-Yeong (2004)

Czechoslovak Mathematical Journal

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We study a generalization of the classical Henstock-Kurzweil integral, known as the strong $ρ$ -integral, introduced by Jarník and Kurzweil. Let $(𝒮_{ρ} (E), ∥ \cdot ∥)$ be the space of all strongly $ρ$ -integrable functions on a multidimensional compact interval $E$ , equipped with the Alexiewicz norm $∥ \cdot ∥$ . We show that each element in the dual space of $(𝒮_{ρ} (E), ∥ \cdot ∥)$ can be represented as a strong $ρ$ -integral. Consequently, we prove that $f g$ is strongly $ρ$ -integrable on $E$ for each strongly $ρ$ -integrable function $f$ if and only if $g$ is...

Displaying similar documents to “Integration and decompositions of weak * -integrable multifunctions”

Displaying similar documents to “Integration and decompositions of weak $^{*}$ -integrable multifunctions”